On the compressible Euler equation and the Langevin deformation of flows on Wasserstein space

Speaker: LI Xiangdong

Venue: 2-312 Haina Court, Zijingang Campus

Abstract: In my previous work and my joint work with Songzi Li, we proved the W-entropy formula for the heat equation of the Witten Laplacian on Riemannian manifolds and the  W-entropy formula for the Wasseestein geodesic flow on Riemannian manifolds. We introduced the Langevin deformation of flows, which is a natural interpolation between the gradient flow of the Boltzmann entropy and the Wasserstein  geodesic flow and is closely related to the compressible Euler equation with damping on manifolds. We proved the existence and uniqueness of the compressible Euler equation with damping on manifolds, the W-entropy formula for the Langevin deformation of flows and its convergence when the viscosity coefficient tends to zero and to infinity respectively. In recent works with Songzi Li, Rong Lei and Yuzhao Wang, we further extend our results to the  compressible $L^p$-Euler equation with damping and the $L^p$-Langevin deformation of flows on over Riemannian  manifolds. Our results are new even for the one dimensional Euler equations with dampings. I will give a survey on these works.